# A Short Proof of Cramér's Theorem in ℝ

@article{Cerf2011ASP, title={A Short Proof of Cram{\'e}r's Theorem in ℝ}, author={Rapha{\"e}l Cerf and Pierre Petit}, journal={The American Mathematical Monthly}, year={2011}, volume={118}, pages={925-931} }

- Published in The American Mathematical Monthly 2011
DOI:10.4169/amer.math.monthly.118.10.925

The law of large numbers asserts that the empirical mean Xn converges (almost surely) towards the theoretical mean E(X1) provided that E(|X1|) is finite. The next fundamental results are the central limit theorem and Cramér’s theorem. Both are refinements of the law of large numbers in two different directions. The central limit theorem describes the random fluctuations of Xn around E(X1). Cramér’s theorem estimates the probability that Xn deviates significantly from E(X1) : P ( Xn > E(X1… CONTINUE READING